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《Chinese Journal of Geophysics》 2015-08
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3-D frequency-domain CSEM modeling using aparallel direct solver

HAN Bo;HU Xiang-Yun;HUANG Yi-Fan;PENG Rong-Hua;LI Jian-Hui;CAI Jian-Chao;Hubei Subsurface Multi-scale Imaging Key Laboratory,Institute of Geophysics and Geomatics,China University of Geosciences;  
Three-dimensional modeling of electromagnetic data is a computationally demanding problem.For frequently-used numerical techniques such as finite-element and finite-difference methods,solving the large linear systems arising from the discretization of Maxwell's equations is a key step which has a major impact on the applicability of the solution,and it has always been a research topic to solve the linear equations efficiently,robustly and accurately.A 3D modeling scheme based on direct solutions of the linear system is presented for frequency domain controlled-source electromagnetic(CSEM)surveys. The Helmholtz equation in terms of secondary electric fields is discretized using a finite-volume(FV)method over a staggered grid.Taking advantage of recent developments in numerical algorithms and the availability of computational resources,the resulting linear system of FV equations is solved directly using the massively parallel solver,namely MUMPS,instead of the most commonly used linear solvers,i.e.Krylov subspace iterative techniques.The direct solver carries out an LU(and possibly LDLT)decomposition of the system matrix and then computes solutions efficiently by applying forward and backward substitutions.To evaluate the computational performance of the direct solver,a series of numerical tests based on synthetic 1D models were conducted,and the results indicate that(1)Normalized residuals of solutions are almost independent of the conductivity value assigned to air layers but increase rapidly as the frequency value decreases.Nevertheless,the order of magnitude of the largest normalized residual is as small as 10-11.At the same time,although the matrix factorization time varies as either the air conductivity or the frequency changes,the variation is only a fraction of the total run time.(2)Both the execution time and required memory increase rapidly(more than linearly)with increasing grid sizes.(3)By executing MUMPS in parallel over multiple processors,not only the total run time but also the average memory used per processor can be reduced a lot.However,the total memory requirement increases with the number of processes.The scalability of MUMPS is limited.Additional numerical experiments considering specific survey settings were done to demonstrate the reliability and effectiveness of the code,and the results are as follows:(1)The FV numerical solutions show excellent agreement with semianalytic solutions for the 1D models.(2)The computation time of a multitransmitter problem is comparable to that of a single-transmitter problem.(3)Reasonable modeling results of 3D models can be obtained for both typical land and marine survey scenarios.In summary,compared with iterative linear solvers,the direct solvers generally benefit CSEM modeling in three aspects.The first is they often provide more accurate solutions.The second is that direct solvers are much more stable for ill-conditioned linear systems,which are almost inevitable because of large electrical conductivity contrasts and/or non-uniform grid.The last is in multitransmitter problems,only a single matrix factorization is necessary,and multiple solutions can be achieved very easily by reusing the factors.The presented 3D CSEM modeling scheme,which employs the MUMPS direct solver,possesses all these advantages.In addition,solving linear systems can be executed in parallel to speed up the computation and to reduce the average memory used per node,although the parallel scalability of MUMPS is limited.In spite of the fact that matrix factorizations for large models can entail tremendous computational cost,it can be anticipated that direct solvers will be used more and more widely as the development of both numerical algorithms and computers.
【Fund】: 国家自然科学基金项目(41274077;41474055);; 中国地质调查局项目(12120113101800)联合资助
【CateGory Index】: P631.325
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